PHYSICAL REVIEW D 94, 025034 (2016) Thermodynamics of perfect fluids from scalar field theory
† ‡ Guillermo Ballesteros,1,2,* Denis Comelli,3, and Luigi Pilo4,5, 1Institut de Physique Théorique, Université Paris Saclay, CEA, CNRS91191 Gif-sur-Yvette, France 2CERN, Theory Division, 1211 Geneva, Switzerland 3INFN, Sezione di Ferrara, 44124 Ferrara, Italy 4Dipartimento di Scienze Fisiche e Chimiche, Università di L’Aquila, I-67010 L’Aquila, Italy 5INFN, Laboratori Nazionali del Gran Sasso, I-67010 Assergi, Italy (Received 1 June 2016; published 27 July 2016) The low-energy dynamics of relativistic continuous media is given by a shift-symmetric effective theory of four scalar fields. These scalars describe the embedding in spacetime of the medium and play the role of Stückelberg fields for spontaneously broken spatial and time translations. Perfect fluids are selected imposing a stronger symmetry group or reducing the field content to a single scalar. We explore the relation between the field theory description of perfect fluids to thermodynamics. By drawing the correspondence between the allowed operators at leading order in derivatives and the thermodynamic variables, we find that a complete thermodynamic picture requires the four Stückelberg fields. We show that thermodynamic stability plus the null-energy condition imply dynamical stability. We also argue that a consistent thermodynamic interpretation is not possible if any of the shift symmetries is explicitly broken.
DOI: 10.1103/PhysRevD.94.025034
I. INTRODUCTION effective field theory (EFT) of continuous media [4,5], which turns to have an ample range of applications. In order Fluid dynamics and thermodynamics are probably the to describe continuous media beyond anisotropic elastic oldest and better known examples of effective descriptions solids, the field content of the pull-back formalism must be of a complicated underlying system in terms of a small extended with a fourth scalar [5]. This allows to include number of macroscopic variables. Systems that admit a superfluids in the picture and also more complex objects fluid description are found in nature at widely separate that are not (yet, maybe) found in nature, such as super- distance scales and energy regimes: from cosmological and solids; see also [6] for other possible types of media. The astrophysical applications to heavy-ion physics and non- fourth scalar can be interpreted as the carrier of an extra relativistic condensed matter. A convenient formulation Uð1Þ charge [5] or as an internal time coordinate of the self- of fluid dynamics in the nondissipative limit is the pull- gravitating medium [7], offering a suggestive link to back formalism—see [1] for a review—where a fluid is massive gravity theories and, in general, models of modi- described through an ensemble of three derivatively fied gravity; see [4,7] and references therein. coupled scalars that are interpreted as comoving coordi- In this work, we focus on perfect fluids, which corre- nates of the fluid’s elements. Within this formalism, fluid spond to two specific subclasses of the EFT of continuous dynamics can be derived from an unconstrained action media at leading order in derivatives (LO), as Fig. 1 principle. A related approach was developed separately illustrates. Although these systems can be considered the to obtain a field theory, symmetry driven, description of the fluctuations—sound waves—propagating in fluids and other types of continuous nonrelativistic media, see [2,3]. The relevant degrees of freedom from this point of view, which we can call phonons, can be identified with the Goldstone bosons of spontaneously broken translational symmetries in the pull-back formalism. Given this, the two approaches can be blended together into a fully relativistic
*[email protected] † [email protected] ‡ [email protected] FIG. 1. Red continuous arrows represent the symmetries (4.2) Published by the American Physical Society under the terms of and (4.7) leading to perfect fluids at leading order in derivatives in the Creative Commons Attribution 3.0 License. Further distri- the EFT of nondissipative continuous media. The blue dashed bution of this work must maintain attribution to the author(s) and arrow indicates that restricting the field content to a single the published article’s title, journal citation, and DOI. (temporal) Stückelberg field leads to (irrotational) perfect fluids.
2470-0010=2016=94(2)=025034(15) 025034-1 Published by the American Physical Society BALLESTEROS, COMELLI, and PILO PHYSICAL REVIEW D 94, 025034 (2016) simplest ones at the level of the energy-momentum tensor, the volume. A simple scaling argument shows that they are not free of subtleties [5], and there is an ongoing s ¼ S=V, the entropy density, is a function of the energy effort towards understanding their properties in depth. Here density, ρ ¼ E=V, and the particle number density, we build upon the work of [5]—see also [4]—where the n ¼ N=V; namely,1 thermodynamic interpretation of effective perfect fluids was studied. We extend the thermodynamic correspond- s ¼ sðρ;nÞ; ð2:1Þ ences proposed in [4,5], obtaining a thermodynamic dic- tionary that we have condensed in Table II. which constitutes the fundamental relation containing Our analysis leads to the conclusion that a general all the thermodynamic information of any simple thermodynamic picture (away from specific limits) requires system. Expressing this relation in the equivalent energy indeed four scalar fields (instead of just three) and, representation, consequently, implies an extension of the pull-back for- malism. Remarkably, the form of the action required for ρ ¼ ρðs; nÞ; ð2:2Þ such a thermodynamic picture is determined by a symmetry group that constitutes a specific set of continuous field and taking its differential, we get the first principle of redefinitions, selecting just two effective operators [5]. thermodynamics: We show that a consistent thermodynamic interpretation requires, in any case, a shift symmetry for each field in the dρ ¼ Tds þ μdn; ð2:3Þ effective action. This is interesting because such symmetry is precisely the minimal requirement to have an EFT where the temperature and the chemical potential are organized as a derivative expansion; see [4,5,8,9]. defined as Moreover, shift symmetries are essential for the under- standing of phonons—the degrees of freedom responsible ∂ρ ∂ρ ≡ μ ≡ for the propagation of sound–as Goldstone bosons [2,3] T ∂ ; ∂ : ð2:4Þ s n n s and also [10]. Finally, we argue that thermodynamic stability of perfect From the additivity of the energy E and the entropy S, the fluids plus the null-energy condition guarantee dynamical Euler relation follows, stability, i.e. the absence of ghost degrees of freedom and of exponential growth of fluctuations (around Minkowski ρ þ p ¼ Ts þ μn; ð2:5Þ spacetime). This holds true for all the types of perfect fluids allowed by the EFT. where p is the intensive variable pressure: Whereas the existence of an effective action description of nondissipative fluid dynamics should not be surprising, ∂E p ¼ − : ð2:6Þ the fact that this action leads to a complete thermodynamic ∂V description of perfect fluids is remarkable and far reaching. S;N Having a unified and general relativistic description of The differential of the Euler relation, together with the first nondissipative dynamics and thermodynamics at the action principle, leads to the fact the intensive variables p, T and μ level may open the possibility for novel applications of the are not independent but satisfy the Gibbs-Duhem relation: pull-back formalism and the effective theory of fluids. dp ¼ sdT þ ndμ: ð2:7Þ II. THERMODYNAMICS IN A NUTSHELL Thermodynamics assumes that the equilibrium states of Given two equations among (2.3), (2.5), and (2.7), the third a simple system can be entirely characterized by the follows. extensive variables volume, V, energy, E, and particle From the definitions of the intensive variables p, T and μ, three equations of state can be written in the energy number of each species, Ni. In addition, it postulates the existence of a function of the extensive variables: the representation: entropy, S, which is maximized in the evolution of the system. These two assumptions constitute the first and T ¼ Tðs; nÞ;p¼ pðs; nÞ; μ ¼ μðs; nÞ: ð2:8Þ second postulates of thermodynamics. The third and fourth postulates establish, respectively, that in a composite All together, these three equations of state are equivalent to system the entropy is an additive function over the the fundamental relation (2.2) or (2.1). Keeping only one or constituent subsystems and that the entropy vanishes at two of them among the three leads to some information zero temperature [11]. For our purposes, it is convenient to use intensive 1In relativistic hydrodynamics, n is usually meant to represent variables, defined by dividing the extensive variables over a charge density.
025034-2 THERMODYNAMICS OF PERFECT FLUIDS FROM SCALAR … PHYSICAL REVIEW D 94, 025034 (2016) TABLE I. Thermodynamic potentials and variables.
Thermodynamic potential Independent variables Legendre transf. to ρ Conjugate variables Energy density ρ s, n none ∂ρ μ ∂ρ T ¼ ∂s jn, ¼ ∂n js Free-energy density F T, n F ρ − Ts − ∂F μ ∂F ¼ s ¼ ∂T jn, ¼ ∂n jT Grand potential density ω T, μω−p ρ − Ts − μn − ∂ω − ∂ω ¼ ¼ s ¼ ∂T jμ, n ¼ ∂μ jT Potential density I s, μ I ρ − μn ∂I − ∂I ¼ T ¼ ∂s jμ, n ¼ ∂μ js about the system. Clearly, the equations of state can also be fluid, the entropy and the particle number currents are both expressed in different ways depending on the two inde- parallel to vμ, and so we write pendent variables that are chosen. In general, given a simple system, two thermodynamic variables among nμ ¼ nvμ;sμ ¼ svμ: ð3:3Þ fs; T; n; μ; ρ;pg can be taken as independent. Starting from the energy representation (2.2), one can By using Eq. (2.5), the perfect fluid EMT can be define other thermodynamic potentials, which are naturally expressed as associated to specific choices of pairs of independent variables different from fs; ng. For instance the Gibbs- Tμν ¼ðTsμ þ μnμÞvν þ pgμν: ð3:4Þ Duhem relation (2.7) already tells us that we can use the pressure as a thermodynamic potential, which corresponds 3 Applying the conservation of the EMT to (3.4), the to (minus) the grand potential (density) ω, i.e. ω ¼ −p.As μ ν projection v ∇ Tμν ¼ 0 leads to (2.7) indicates, μ and T are the associated independent variables in this case. We can also introduce the free-energy ∇α μ∇α 0 density F ¼ ρ − Ts, which, using (2.3) and (2.5), satisfies T sα þ nα ¼ ; ð3:5Þ dF ¼ μdn − sdT, effectively selecting n and T. Similarly, we define another potential density—which bears no where we have used the Gibbs-Duhem relation (2.7) along 0 μ0 0 0 μ∇ standard name—I ¼ ρ − μn so dI ¼ Tds − ndμ. These the fluid flow p ¼ n þ sT , denoting by f ¼ v μf the are all the potentials we will need for our purposes. The Lie derivative of a function f along the four-velocity of the relations among them are summarized in Table I. fluid. The equation (3.5) is equivalent to the first principle of thermodynamics, and it tells us that for a perfect fluid μ III. ENERGY-MOMENTUM TENSOR AND (with nonvanishing chemical potential, ) a change in the CONSERVED CURRENTS OF PERFECT FLUIDS particle current implies a variation of the entropy current. If the particle number current is conserved (or if μ is The energy momentum tensor (EMT), Tμν, and the negligible), the entropy current is also conserved and we 0 μ currents of a macroscopically continuous system play a can write s þ θs ¼ 0, where θ ¼ ∇ vμ is the expansion. In μ 0 pivotal role in the determination of its thermodynamic particular, if ∇ nμ ¼ 0, the equation n þ θn ¼ 0 also interpretation (if any). Of particular relevance are the σ μ μ holds and we find that the entropy per particle ¼ s=n entropy and particle currents, s and n , respectively. In is conserved along the flow lines: σ0 ¼ 0, thus defining an agreement with the second postulate of thermodynamics, adiabatic fluid. Instead, an isentropic fluid is defined as one that has constant entropy per particle, i.e. ∇μσ ¼ 0. ∇ μ ≥ 0 μs ; ð3:1Þ Using hμν ¼ gμν þ vμvν to project the conservation of the EMT orthogonally to uμ, we get the Euler equations for where the equality applies only for reversible (nondissipa- a perfect fluid, tive) processes. In this work, we focus on perfect fluids, which by ρ ν∇ 0 the definition are the continuous media whose energy- ðp þ Þaμ þ hμ νp ¼ ; ð3:6Þ momentum tensor is of the form where ðp þ ρÞ=n is the enthalpy per particle and aμ ¼ μ ν∇ Tμν ¼ðρ þ pÞvμvν þ pgμν;vvμ ¼ −1; ð3:2Þ v νvμ is the acceleration along the flow lines. This equation manifestly shows that the acceleration aμ depends where the four-velocity of the fluid, vμ, is the (unique) on the pressure gradient, as expected, since ρ þ p is the 2 timelike eigenvector with unit norm of Tμν. For a perfect relativistic generalization of the mass density.
2Throughout the paper, we use the metric signature 3We thank S. Matarrese for pointing out the paper [10] in ð−; þ; þ; þÞ. which a similar treatment to the one in this section is given.
025034-3 BALLESTEROS, COMELLI, and PILO PHYSICAL REVIEW D 94, 025034 (2016) Defining the vorticity tensor as (see e.g. [12]) We require that the action of these four fields respects the shift symmetries Ωμν ∇νwμ − ∇μwν ; 3:7 ¼ð Þ ð Þ A A A A Φ → Φ þ f ; ∂μf ¼ 0;A;μ ¼ 0; 1; 2; 3: ð4:1Þ μ ρ μ from the enthalpy current w ¼ð þ pÞ=nv , we can In addition, we impose invariance under internal spatial express the conservation of the EMT in terms of the volume preserving diffeomorphisms Carter-Lichnerowicz equations: ∂Ψa a a b ν ν VsDiff∶ Φ → Ψ ðΦ Þ; det ¼ 1; nΩανv ¼ nT∇ασ − wα∇ nν: ð3:8Þ ∂Φb a; b ¼ 1; 2; 3: ð4:2Þ Indeed, (3.5) and (3.6) come from projecting (3.8) along μ v and orthogonally to it. With these symmetries, at leading order in the derivative A perfect fluid is sometimes said to be irrotational if expansion (LO), there are only two timelike independent Ωμν ¼ 0. In this sense, an irrotational perfect fluid with four-vectors and just three independent scalar operators, μ ∇ nμ ¼ 0 satisfies ∇μσ ¼ 0 and is called isentropic. which can be chosen to be Clearly, a fluid that is adiabatic (σ0 ¼ 0) is not necessarily ϵμαβγ ∂μΦ0 irrotational in the previous sense. μ a b c μ ffiffiffiffiffiffiffi u ¼ − pffiffiffiffiffiffi ϵabc∂αΦ ∂βΦ ∂γΦ ; V ¼ − p It is also common usage to call irrotational a perfect fluid 6b −g −X whose four-velocity is the derivative of a scalar quantity, ð4:3Þ i.e. vμ ¼ ∂μΨ. Clearly, this notion is not equivalent, in general, to Ωμν 0. ¼ and ffiffiffiffiffiffiffiffiffiffi p 0 μ 0 μ 0 IV. EFFECTIVE ACTION FOR NONDISSIPATIVE b ¼ det B;X¼ ∂μΦ ∂ Φ ;Y¼ u ∂μΦ ; ð4:4Þ HYDRODYNAMICS where the four-vectors have norm −1 and B is the 3 × 3 The study of relativistic fluid dynamics from an uncon- ab μ a b matrix of components B ¼ ∂ Φ ∂μΦ . Then, the LO strained action principle has a long history. In this work, we 5 will closely follow the treatment of the subject that we gave action including gravity is [5,16] (see also [7]) Z Z in [7]. Our approach is related to Carter’s geometrical ffiffiffiffiffiffi ffiffiffiffiffiffi 2 4 p− 4 p− formulation of the problem [13], but embedded in an S ¼ Mpl d x gR þ d x gUðb; Y; XÞ; ð4:5Þ effective field theory (EFT) framework; as proposed e.g. in [2,3] and later in [4]; see also [5]. Various aspects and 2 where M ¼ 1=ð16πGÞ, with G being Newton’s constant. applications of the EFT approach for describing conti- pl nuous media have been developed in [4–6,8,9,14–23]. The gravitational EMT tensor is Interestingly, this framework can be used, for instance, Tμν U − bU gμν YU − bU uμuν 2XU VμVν; to treat massive and modified gravity in a unified way, ¼ð bÞ þð Y bÞ þ X interpreting them as self-gravitating media; see [7].4 For the ð4:6Þ development of Carter’s idea into the subsequent pull-back formalism, see [27,28]. A review of this formalism is given where U is an arbitrary master function and we have μ in [1] and recent applications in the context of cosmology denoted by Ub, UX, and UY its partial derivatives. Since u can be found in [7,9,17,18,21,29–34]. The idea that lies at and Vμ are, in general, not parallel, this EMT does not the core of this treatment consists in using the Lagrangian describe a perfect fluid like (3.2). Actually, (4.6) was coordinates of a continuous medium as the low-energy proposed as the EMT of a superfluid; see [35,36] (and also degrees of freedom of the EFT. In the original pull-back e.g. [16,28] within the framework we use). Indeed, since a 0 formalism, three scalar fields Φ , a ¼ 1, 2, 3, identify the Vμ ∼ ∂μΦ is irrotational, it has been associated to the fluid elements as they propagate in space. Here we add a intrinsic superfluid component of the medium, whereas uμ fourth scalar Φ0, which may be interpreted at this stage as would correspond to the standard fluid component. an internal time coordinate of the medium. Indeed, these In this work, we will be solely interested in perfect fluids, scalars can be seen as Stückelberg fields restoring broken which can be obtained either from uμ or Vμ using the action diffeomorphisms in four-dimensional spacetimes; see e.g. (4.5), as the EMT (4.6) shows neatly. Concretely, if U ¼ [7,24–26]. Uðb; YÞ or U ¼ UðXÞ, the resulting medium is a perfect
4See also [24–26] for previous works on massive gravity using 5As explained in [9], the Einstein-Hilbert term is the unique the Stückelberg “trick.” possible choice at this order in derivatives.
025034-4 THERMODYNAMICS OF PERFECT FLUIDS FROM SCALAR … PHYSICAL REVIEW D 94, 025034 (2016) fluid. Remarkably, the first of these two cases is obtained (iv) Uðb; YÞ has the same currents as UðYÞ because they requiring that the action has to be invariant under [5] (see share field content and symmetries. We recall that also e.g. [7,8,16]) Uðb; YÞ is the most general case at LO assuming that the action is symmetric under VsDiff and Ts. 0 0 a μ μ μ Ts∶ Φ → Φ þ fðΦ Þ;a¼ 1; 2; 3; ð4:7Þ As we will see in Sec. V, only the currents J , Y , and X are needed to establish the full thermodynamic dictionary. with f being an arbitrary function. Clearly, UðbÞ and UðYÞ Actually, other conserved currents are present (depending also describe perfect fluids. Assuming that the action on the operator content) but are not relevant for our depends only on the spatial Stückelbergs Φa, a ¼ 1,2, purposes in this work; for a discussion, see for instance 3, the only possibility respecting VsDiff is UðbÞ. Similarly, [4,5,7,15,18]. Requiring only the symmetry VsDiff and the 0 UðXÞ is the only possibility if the action contains only Φ shift symmetry (4.1) for Φ0 the action is (4.5) and the EMT and respects a shift symmetry. These two types of perfect is not a perfect fluid. In this case, the conserved current fluids, UðbÞ and UðXÞ, are manifestly different from each associated to the second of these symmetries is precisely μ other since V is irrotational—in the sense that it is the the sum of the currents (4.9) and (4.10): X μ þ Yμ.As 0 μ gradient of the scalar Φ —and u is not. before, any current of the form ðX μ þ YμÞfðΦaÞ is also μ a At higher orders in the derivative expansion, the conserved, by virtue of u ∂μΦ ¼ 0. symmetries we have discussed do not protect the perfect fluid form of the EMT of these systems, which generically B. Energy density and pressure of perfect fluids acquires other terms; see e.g. [5,9,19]. This means that from the point of view of the EFT, the “perfectness” of perfect Barotropic fluids are common in several branches of fluids is only an approximate low-energy feature. physics; in particular in cosmology. They serve to model in a first and crude approximation the basic matter species of A. Conserved currents of effective perfect fluids the ΛCDM model: cold dark matter (CDM), baryons, The different kinds of perfect fluids that can be obtained photons and neutrinos. The current accelerated expansion out of the general EMT (4.6) can be classified according to can also be modeled with a barotropic fluid, be it a cosmological constant, Λ, or a more exotic component their conserved currents; see also [7]. These currents are of ρ two types: Noether currents and currents that are conserved with a sufficiently negative equation of state w ¼ p= . The independently of any symmetry of the action. In what usual definition is that barotropic fluids are those whose follows when we say that a current is conserved we mean pressure is a function of the energy density alone, i.e. μ p ¼ pðρÞ. This can be generalized by defining a barotropic that it is covariantly conserved, i.e. ∇μJ ¼ 0. (i) UðbÞ has currents of two types. First, there is fluid as one whose pressure is completely characterized knowing the energy density at each point of the fluid (or J μ ¼ buμ; ð4:8Þ vice versa) [18]. In this sense, the perfect fluids UðXÞ, UðbÞ, and UðYÞ are all barotropic. Indeed, in these cases, which is conserved off-shell. Actually, any current the pressure and the energy density are related, respectively, of the form fðΦaÞJ μ, where f is a function of the through the relations a μ a fields Φ , is also conserved thanks to u ∂μΦ ¼ 0. ρ − − This last equation shows that Φa are actual comov- ðiÞ ¼ UðbÞ;p¼ U bUb;
ing coordinates of the fluid, as the pull-back for- ðiiÞ p ¼ UðXÞ; ρ ¼ 2XUX − U; malism requires. ρ − (ii) UðXÞ has only one independent conserved current, ðiiiÞ p ¼ UðYÞ; ¼ YUY U: ð4:11Þ ffiffiffiffiffiffiffi μ p μ The perfect fluids Uðb; YÞ are not barotropic, in general, X ¼ −2 −XU V ; ð4:9Þ X simply because their Lagrangians are functionals of two 0 0 0 0 independent operators: coming from the symmetry Φ → Φ þ c , ∂μc ¼ 0, Φ0 and giving the dynamics of for UðXÞ. The factor ðivÞ p ¼ U − bU and ρ ¼ YU − U; U ¼ Uðb; YÞ: 2 is included for later convenience. b Y (iii) UðYÞ has, in addition to the same currents as UðbÞ, ð4:12Þ the current The relation between pressure and energy density for U ¼ μ μ Y ¼ UYu : ð4:10Þ Uðb; YÞ can be determined only if another thermodynamic quantity is also known. However, there exist specific As a matter of fact, any fðΦaÞYμ is conserved as choices of the function Uðb; YÞ that are barotropic. In well. These currents are due to the symmetry particular, a constant w ¼ p=ρ can be obtained for 0 0 a 1þw −w Ts: Φ → Φ þ fðΦ Þ. Uðb; YÞ¼b UðYb Þ, with U being an arbitrary
025034-5 BALLESTEROS, COMELLI, and PILO PHYSICAL REVIEW D 94, 025034 (2016)
1þw 1þ1=w function. Another possibility is Uðb;YÞ¼b þ λY , s ∝ UY and μ ∝ Ub; see Table II. An analogous procedure with constant λ. A constant equation of state can also be can be followed when the thermodynamical variables ðs; μÞ reproduced with UðXÞ¼Xð1þ1=wÞ=2. Nonrelativistic matter are used, in this case the natural thermodynamic potential is with p ¼ 0, such as standard CDM, admits only the I and one can check that setting Uðbðs; μÞ;Yðs; μÞÞ ¼ description UðbÞ¼b. −Iðs; μÞ all the thermodynamic relations are satisfied if There are also other examples of perfect barotropic fluids b ¼ s and Y ¼ T; the result is also given in Table II. The that have been used often in astrophysics and cosmology, possibility of identifying −Uðb; YÞ as the thermodynamic and can be easily described within our framework. For potential Iðs; μÞ was already found in [5]. The dictionary of Table II shows that some combinations instance, a Chaplygin gas satisfies p ¼ −A=ρ, withffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi con- p 2 of operator content and independent thermodynamic var- stant A, and can be obtained from UðbÞ¼ A þ λb , pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ 2 λ λ iables have no entry. By looking at Table II, for the case UðYÞ¼ A þ Y or UðXÞ¼ A þ X, with constant . Uðb; YÞ, note that in the dictionary there is no (direct) entry for some choice of independent thermodynamical varia- V. THERMODYNAMIC DICTIONARY FOR bles. This is the case for the choice of variables ðμ;TÞ and PERFECT FLUIDS ðn; sÞ for Uðb; YÞ. The reason is that the EMT tensor for −ω ρ In this section, we give the thermodynamic interpretation Uðb; YÞ is such that U is not proportional to p ¼ or to U of the EFT of perfect fluids described in Sec. IV by giving as thermodynamics should require (see Table I)if is the correspondence between the thermodynamic variables interpreted to be the corresponding thermodynamical potential. As it was already argued in [5], U b; Y can and the operators b, Y, and X for the different kinds of ð Þ be identified with −I s; μ because this potential can be perfect fluids. These results extend the interpretations that ð Þ obtained from ρ n; s using a Legendre transformation; see were previously proposed in [4,5]. ð Þ Table I. The identification of U b; Y with −F n; T can be For a simple thermodynamic system, among the six ð Þ ð Þ argued on the same ground. variables (s, T, n, μ, ρ, p), two of them can be taken as It is important to stress that the nonexistence of an entry independent, say z1 and z2, and the remaining four can be in the dictionary for some variables does not mean that it is expressed in terms of z1 and z2. The EMT provides the impossible to use those variable. Take, for instance, the explicit form of p and ρ; see (4.11)–(4.12). Our outcome, entries relative to Uðb; YÞ and variables ðn; TÞ. By using a once the independent variables have been chosen, is that U Legendre transformation, one can always switch from is proportional to the natural thermodynamical potential ðn; TÞ to ðn; sÞ, still getting T ¼ Y and μ ¼ −U . Thus, associated with such thermodynamical variables; see b the entries in the dictionary correspond to inequivalent Table I. In the Appendix, we describe a method based thermodynamic interpretations of the very same scalar field on the counting of derivatives of U with respect to the EFT action. operators, showing that this is indeed the case. Let us now consider the Lagrangian UðbÞ. In this case Let us start with the fluid described by Uðb; YÞ, which given the fact that at least two independent thermodynam- depends on two operators, and choose as independent ical variables are needed to describe a simple system and a thermodynamic variables ðT;nÞ. From (4.12), we have that single operator is present in the action, it is not surprising ρ ¼ YUY − U and p ¼ U − bUb. If a thermodynamic interpretation exists, the operators b and Y of the scalar that various possibilities exist. Using the Euler relation field theory should be functions of the chosen thermody- (2.5) and the expressions (4.11) for the energy density and μ − namic variables, namely, Y ¼ YðT;nÞ and b ¼ bðT;nÞ. pressure, we get n þ Ts ¼ bUb. In addition, the Gibbs- μ − From Table I, it is clear that the natural thermodynamic Duhem equation (2.7) tells us that sdT þ nd ¼ bUbbdb, potential with variables ðT;nÞ is the free-energy density F, and the first principle of thermodynamics (2.3) is μ − which is obtained from ρ or p after a single Legendre Tds þ dn ¼ Ubdb. To proceed further, we can choose transformation. The natural choice is to take Uðbðn; TÞ; n and T as our two independent thermodynamic variables, Yðn; TÞÞ ¼ −Fðn; TÞ and find for which b ¼ bðn; TÞ and so that b ¼ bðn; TÞ. It is easy to check that n ¼ b and μ ¼ − Y ¼ Yðn; TÞ the fundamental thermodynamic relations Ub solve these equations for all Ub. This implies that (2.3) and (2.5) are satisfied. There are multiple ways in s ¼ 0 and, according to the fourth postulate of Sec. II, which this can be done, all of them leading to the same set T ¼ 0. Therefore, UðbÞ can describe a perfect fluid in the of solutions. Using that dF ¼ μdn − sdT—see Table I— limit of zero temperature. With this interpretation, that was J μ the Euler relation (2.5) becomes bF b − YF Y ¼ nF n − already proposed in [4], the current defined in (4.8) TF T. Expressing the derivatives with respect to b and n plays the role of the conserved particle current, and the in terms of derivatives with respect to T and n and imposing entropy current vanishes in this limit. that the resulting equation has to be valid for all F,we Let us consider now as independent thermodynamic obtain b ¼ nbn − TbT and Y ¼ TYT − nYn. We can now variables ðμ;TÞ, i.e. we assume b ¼ bðμ;TÞ. It is easy to use the Gibbs-Duhem relation (2.7) or, simply, the defi- see that b ¼ s and T ¼ −Ub can solve the thermodynamic nition F ¼ ρ − Ts. Either way, we are led to T ∝ Y, n ∝ b, equations. For consistency, this implies that n must be zero.
025034-6 THERMODYNAMICS OF PERFECT FLUIDS FROM SCALAR … PHYSICAL REVIEW D 94, 025034 (2016) TABLE II. The EFT-thermodynamics dictionary for perfect fluids. The first row gives the energy density, the pressure and the relevant currents for the thermodynamic interpretation of each type of perfect fluid. The other rows give the entries of the dictionary for the four pairs of independent thermodynamic variables and their corresponding potentials. When it appears, f is an undetermined function (and f0 is its derivative) of the argument z ¼ μ=T or σ ¼ s=n.
UðbÞ UðYÞ UðXÞ Uðb; YÞ ρ ¼ −U ρ ¼ −U þ YUY ρ ¼ −U þ 2XUX ρ ¼ −U þ YUY p ¼ U − bUb p ¼ U p ¼ U p ¼ U − bUb J Y 1=2 J Y μ ¼ buμ μ ¼ UY uμ X μ ¼ −2ð−XÞ UXVμ μ ¼ buμ, μ ¼ UY uμ μ s μ 2 μ ð ; Þ b ¼ s Y ¼ X ¼ −μ ffiffiffiffiffiffiffi b ¼ s, Y ¼ I − 0 p ¼ U n ¼ n ¼ UY n ¼ −2U −X n ¼ UY − 0 X − T ¼ Ub T ¼ T ¼ 0, X μ ¼ −n=2Vμ T ¼ Ub J Y J Y μ ¼ suμ μ ¼ nuμ X μ ¼ nVμ μ ¼ suμ, μ ¼ nuμ n T 2 ð ; Þ b ¼ n Y ¼ T X ¼ −T ffiffiffiffiffiffiffi b ¼ n, Y ¼ T F − 0 p ¼ U s ¼ s ¼ UY s ¼ −2U −X s ¼ UY μ − μ 0 X μ − ¼ Ub ¼ μ ¼ 0 ¼ Ub J Y J Y μ ¼ nuμ μ ¼ suμ X μ ¼ sVμ μ ¼ nuμ; μ ¼ suμ ðμ; TÞ Y ¼ TfðzÞ X ¼ −T2fðzÞ ω − − 0 0 ¼ U s ¼ UY ðf zf Þ s ¼ UXðμf − 2TfÞ μ 0 0 z ¼ =T n ¼ f UY n ¼ −UXTf Y ρ 1=2 Tf μ ¼ð þ pÞuμ Tf X μ ¼ðρ þ pÞVμ ðn; sÞ b ¼ sfðσ−1Þ ρ − 0 ¼ U μ ¼ −Ubf σ 0 ¼ s=n T ¼ −Ubðf − f =σÞ J μ ¼ sfuμ
According to the discussion in Sec. II, the choice n ¼ 0 addition, the limits for ðμ;sÞ and ðn; TÞ can be obtained should be understood as zero particle density. However, in choosing specific functions f for the fluids that depend on a relativistic hydrodynamics, n is usually meant to represent single operator. Concretely, this can be done choosing f to a charge density, making this point of view more appealing. be a constant or, imposing f ¼ n=s for UðbÞ; f ¼ μ=T for μ In this case, b ¼ s and the current J represents the UðYÞ and f ¼ μ2=T2 for UðXÞ. entropy current. This is the interpretation that was earlier The analysis of this section leads to the conclusion that advocated in [5] and subsequent works. Uðb; YÞ arises as the most appropriate Lagrangian for a It is interesting to note that there are choices of complete thermodynamic description of a relativistic perfect independent thermodynamic variables for which an arbi- fluid. Having two effective operators allows a full matching trary function of their ratio appears in the dictionaries of to the thermodynamic relations describing a simple UðbÞ. Consider, as an example, UðbÞ and take n and s as system. This is translated into the fact that for Uðb; YÞ, independent variables, so that the natural thermodynamic the currents J μ and Yμ can be put in correspondence with potential is the energy density, ρðn; sÞ. The equation ρ þ the entropy and particle currents—both of which are con- U ¼ 0 is automatically satisfied, whereas the Euler relation served—giving an adiabatic fluid; see Sec. II.The becomes b ¼ nbn þ sbs, whose general solution is Lagrangian Uðb; YÞ was already identified in [5] as the b ¼ sfðn=sÞ, with f being an unspecified function of most general one for a perfect fluid carrying a conserved σ−1 ¼ n=s. The conservation of the current J μ ¼ buμ charge. Moreover, it is remarkable that Uðb; YÞ is selected gives ðρ þ pÞθ ¼ −Ts0 − sT0, which is the first principle from the EFT of continuous media at LO in derivatives by of thermodynamics in the form (3.5). Similarly, for UðXÞ the symmetry VsDiff ×Ts;seeFig.1. In addition, thanks to and UðYÞ with μ and T as independent thermodynamic this symmetry, Uðb; YÞ includes all four Stückelberg fields variables, the corresponding entries of the dictionary needed for a full embedding of the fluid in spacetime, but depend on the ratio z ¼ μ=T. An interesting discussion without introducing two different four-vectors; see Sec. IV of the case UðXÞ can be found in [10]. When an unspecified and also [7]. It is also worth pointing out that Uðb; YÞ is a function f is found, it also enters in the conserved currents nonbarotropic (and nonisentropic) fluid, which opens the and their physical interpretation is more subtle. possibility of describing the dynamics and thermodynamics Notice that the entries for UðbÞ and UðYÞ relative to the of a broad variety of physical systems. In summary, for a variables ðμ;sÞ and ðn; TÞ represent also the limiting cases complete thermodynamic description of perfect fluids at low of Uðb; YÞ for UY ¼ 0 and Ub ¼ 0, respectively. In energies as simple systems, the standard pull-back formalism
025034-7 BALLESTEROS, COMELLI, and PILO PHYSICAL REVIEW D 94, 025034 (2016) must be extended with an extra scalar, Φ0, and with the degeneracy g per energy state. The number of particles per symmetry VsDiff × Ts. unit of momentum k at temperature T is a function of the chemical potential μ: VI. THERMODYNAMICS WITH BROKEN g SHIFT SYMMETRY? 2π2Fðk; T; μÞ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ϵ ¼ 1: exp ½ð k2 þ m2 − μÞ=T þϵ In this section, we argue that the shift symmetries (4.1) are essential for a consistent thermodynamic interpretation ð7:1Þ of the EFT of perfect fluids. Generically, it is sufficient that the action of the four Stückelberg fields does not respect In this expression, ϵ ¼ 1 corresponds to FD and ϵ ¼ −1 to one of these symmetries to prevent a proper thermody- BE. For convenience, we have set the Boltzmann constant namic description. To illustrate this point, we will consider to be 1. Some interesting limits are controlled by the ratios the simplest possible example, assuming that the shift m=T and z ¼ μ=T.Ifz ≫ 1, the gas becomes degenerate; symmetry (4.1) of the case UðXÞ is broken by the explicit whereas, if z ≪ 1 the particles have more freedom to appearance of Φ0 on the master function. So, we consider a occupy higher energy levels. The deeply relativistic limit is 0 ≫ k-essence Lagrangian [37–40], given by UðX; Φ Þ. The T m and, conversely, the gas becomes nonrelativistic at μ sufficiently low temperatures, i.e. m ≫ T. (nonvanishing)ffiffiffiffiffiffiffi covariant divergence of the current X ¼ p μ 0 For T ≫ m, we get −2 −XUXV is the equation of motion for Φ , i.e. μ 0 0 ∇ X μ ¼ U , where U ¼ ∂Φ0 U. The EMT is formally of ρ −ϵ g 4 −ϵα −ϵ g 3 −ϵα the same form as for UðXÞ, thus describing a perfect fluid p ¼ 3 ¼ π2 T Li4ð Þ;n¼ π2 T Li3ð Þ; ρ 2 − V with ¼ XUX U, p ¼ U and four-velocity μ. From 4 ρ the relations (3.4) and (3.5), we find s ¼ − n logðαÞ; ð7:2Þ 3 T pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi ν ν 0 μ sν∇ T þ nν∇ μ ¼ −XU þ X ∇μ −X: ð6:1Þ where LixðzÞ is the polylogarithm function of order x [41] and α ¼ expðzÞ is the fugacity. Using Table II, we can Let us now attempt to construct a thermodynamic inter- reproduce these relations by choosing ðμ;TÞ and taking pretation of k-essence by considering the various possible μ combinations of two independent variables among n, T, ϵ 1=4 g 4 z U Y Y ;YTf z T − 4 −ϵe : and s, in the same way as in the previous section: ð Þ¼3 ¼ ð Þ¼ π2 Li ð Þ (i) ðμ;sÞ: In this case we get X ¼ −μ2 and Φ0 is independent from μ and s. We find that T ¼ 0 ð7:3Þ and X μ ¼ nuμ, so that (6.1) implies that U0 ¼ 0. (ii) ðn; TÞ: Then X ¼ −T2 and Φ0 is independent from n The high temperature limit of a deeply relativistic gas is and T. Besides, μ ¼ 0 and X μ ¼ suμ. Again, (6.1) obtained taking α → 0, which gives Y ∝ T. In the low- implies that U0 ¼ 0. temperature limit, α → ∞, the gas is degenerate and (iii) ðμ;TÞ: Taking UðXðμ;TÞ¼pðμ;TÞ, from the −ϵα → αn ∝ μ Linð Þ nϵ. As a result, Y . The same conclusions Euler and the Gibbs-Duhem relations, we obtain 2 3 2 can be reached by using instead U ¼ gX = , with X ¼ that X ¼ −μ f1ðT=μÞ and ϕ ¼ f2ðT=μÞ with 2 α α −ϵ −ϵα 1=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi T f2ð Þ and f2ð Þ¼½π2 Li4ð Þ . 0 0 0 s ¼ −f U −X=f1 þ f U = −X=f1 and n ¼ 1 pXffiffiffiffiffiffiffiffiffiffiffiffi 2 In the nonrelativistic limit m ≫ T, the expressions (7.1) 0 − 2 − 0 0 ðf1T Xf1ÞUX þ f2f1TU =X. Since the cor- become Φ0 respondence must be valid for all UðX; Þ, the equation (6.1) implies that U0 ¼ 0. mT 3=2 μ − m 3T n ¼ g exp ; ρ ¼ n m þ ; The thermodynamic relations are incompatible with the 2π T 2 equations of motion unless the Lagrangian does not depend ≪ ρ on Φ0. We conclude that k-essence only admits a thermo- p ¼ nT : ð7:4Þ dynamic interpretation if the shift symmetry is enforced, that is: if the action depends only on X. The above These can be reproduced with Uðb; YÞ, where b ¼ n, statement can be easily extended to the other cases. Y ¼ T, and taking g mY 3=2 VII. SOME SIMPLE APPLICATIONS OF THE Uðb; YÞ¼bY 1 þ Log − bm: ð7:5Þ THERMODYNAMIC DICTIONARY b 2π
A. Bose-Einstein and Fermi-Dirac distributions When both α and m=T are non-negligible, getting Consider the Bose-Einstein (BE) and Fermi-Dirac (FD) a relation among the EFT operators b and Y and statistics for (noninteracting) particles of mass m with spin the thermodynamic variables involves solving an
025034-8 THERMODYNAMICS OF PERFECT FLUIDS FROM SCALAR … PHYSICAL REVIEW D 94, 025034 (2016) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi integrodifferential equation for Uðb; YÞ, using b ¼ nðT;μÞ μ ν T −gμνξ ξ ¼ constant: ð7:8Þ and μ ¼ −Ub þ YUbY. An analogous relation holds as well for the chemical B. van der Waals gas potential μ; see for instance [42]. The temperature T and In this case the pressure and the particle number density the chemical potential μ of our dictionary—see Table II— are related by ðp þ κn2Þð1 − γnÞ¼Tn, where the con- are consistent with the Ehrenfest-Tolman result provided μ μ μ 0 stants κ and γ model the finite molecular volume of the gas that v ∝ ξ and ξ ∂μΦ ¼ 1. Indeed, T and μ can be and small intermolecular interactions, respectively. This associated to Y and −X2, respectively. For a static space- equation of state can be easily obtained from Uðb; YÞ time, the timelike Killing vector ξμ can be identified with choosing the pair fn; Tg as independent thermodynamic ∂μΦ0 and thus (7.8) holds, showing that the Ehrenfest- variables, so b ¼ n and Y ¼ T. Integrating p ¼ U − bUb Tolman effect takes place. we obtain U ¼ b½bκ þ Y log ðb−1 − γÞþUðYÞ . The func- tion UðYÞ can be determined imposing that for κ ¼ 0 and VIII. THERMODYNAMIC STABILITY γ ¼ 0 we recover the expression (7.5) for the ideal gas, which leads to Thermodynamic stability of a system requires that the Hessian matrix of the function sðρ;nÞ must be negative 6 gð1 − γbÞ mY 3=2 definite, which is guaranteed by the conditions Uðb; YÞ¼b bκ þ Y − m þ Y log ; b 2π 2 sρρ þ snn ≤ 0 and sρρsnn − sρn ≥ 0: ð8:1Þ ð7:6Þ These equations can also be equivalently formulated in and hence terms of conditions for the energy density, whose Hessian must be positive definite: 3 ρ ¼ b m þ T − nκ ; ρ ρ ≥ 0 ρ ρ − ρ2 ≥ 0 2 ss þ nn and ss nn sn : ð8:2Þ 5b mY 3=2 gð1 − bγÞ s ¼ þ b log ; It is also possible to use the other potentials of Table I.In 2 2π b particular, for the free-energy density Fðn; TÞ¼ρ − Ts, F ≥ 0 F ≤ 0 bYγ b mY −3=2 the conditions reduce simply to nn and TT μ ¼ m − 2bκ þ þ Y log : (while the condition on the mixed second derivative is 1 − bγ gð1 − bγÞ 2π automatically implied by these two). Similarly, for the ð7:7Þ potential Iðμ;sÞ¼ρ − μn the conditions are Iμμ ≤ 0 and Iss ≥ 0. If any of these potentials (F or I) are identified − C. Polytropic fluids with U, which according to Table II occurs only for Uðb; YÞ, the thermodynamic stability conditions are trans- Polytropic fluids are used to model the behavior of lated into simple constraints on the derivatives of U. matter under a wide range of physical conditions, includ- Concretely: ing, e.g., the interior of neutron stars. It is convenient to separate the mass contribution ρ0 ¼ nm (m is the individual Ubb ≤ 0 and UYY ≥ 0: ð8:3Þ mass of the fluid’s constituents) and the internal energy ϵ ρ ρ ϵ density I from the energy density ¼ 0 þ I. The Similarly, for the grand potential ωðμ;TÞ¼−p ¼ Γ equation of state then reads p ¼ κρ0 , with Γ constant. ρ − sT − nμ, thermodynamic stability is guaranteed when A polytropic equation of state can be described by bΓ ω ω ≤ 0 ω ω − ω2 ≥ 0 UðbÞ¼λb þ κ 1−Γ, taking fn; Tg as thermodynamic var- μμ þ TT ; μμ TT μT : ð8:4Þ iables. In this case, b ¼ n and ρ0 ¼ bm, pðbÞ¼ Γ If − U is identified with ρ or ω, as it is the case for the U − bUb ¼ κb . Polytropic equations of state can also be described by UðYÞ and UðXÞ, though the expressions barotropic fluids that depend only on X, b or Y, the are more involved and will be omitted. thermodynamic dictionary involves an undetermined func- tion, f, of a ratio of independent thermodynamic variables; see Table II. In these cases, thermodynamic stability leads D. Ehrenfest-Tolman effect not only to conditions on the derivatives of U, but also to If the spacetime curvature is nonzero and there exists a ξμ timelike Killing vector , the equilibrium temperature T 6A 2 × 2 matrix M is negative definite if TrðMÞ ≤ 0 and satisfies—according to the Ehrenfest-Tolman effect—the detðMÞ ≥ 0. Conversely, it is positive definite if TrðMÞ ≥ 0 and relation detðMÞ ≥ 0.
025034-9 BALLESTEROS, COMELLI, and PILO PHYSICAL REVIEW D 94, 025034 (2016) some constraints involving derivatives of f. Concretely, for ∂p U − bU ∂p bU c2 Y bY ;c2 bb : ρ ¼ −UðbÞ we get b ¼ ∂ρ ¼ Y ¼ ∂ρ ¼ − b YUYY Y Ub YUbY
00 ð9:3Þ Ubb ≤ 0 and f Ub ≤ 0; ð8:5Þ To obtain the relation between the variations of the where primes denote derivatives of f with respect to its energy density and the pressure we make use of the argument; see Table II. Similarly, for ω ¼ −UðYÞ we μ μ μ μ conservation of the currents J ¼ bu and Y ¼ UYu , obtain defined in (4.8) and (4.10). This can be written as b0 þ 0 0 μ θb ¼ 0 and U b þ U Y þ θU ¼ 0 , where θ ¼ ∇μu ≥ 0 00 ≥ 0 Yb YY Y UYY and f UY : ð8:6Þ is the expansion, that we introduced in Sec. III. Combining these two equations to eliminate θ, we get Finally, in the case ω ¼ −UðXÞ, we get 0 2 0 bY ¼ cbYb : ð9:4Þ 0 2 00 2XUXX þ UX ≤ 0; and ððf Þ − 2ff ÞUX ≥ 0: ð8:7Þ Using this result, the sound speed (9.1) can be easily If we take the case of the free relativistic BE or FD computed as (see also [16]) distribution (see Sec. VII), we can specify the function μ 00 ≥ 0 0 0 4 ρ fðzÞ, where z ¼ =T, and then we get: f ðzÞ when 2 pbb þ pYY cbY Y þ bpb ∼ −ϵ −ϵ z 1=4 cs ¼ 0 0 ¼ ; ð9:5Þ f ð Li4ð e ÞÞ (which is the case for the UðYÞ ρbb þ ρYY ρ þ p 0 2 00 and it implies UY ≥ 0) while ððf Þ − 2ff Þ ≤ 0 when f ∼ z 1=2 ð−ϵLi4ð−ϵe ÞÞ (for the UðXÞ case and it implies where, as we have been doing through, the subscripts in 7 U ρ p UX ≤ 0). , and denote partial derivatives, e.g. ∂ρ ∂ ρ p − IX. SOUND SPEED Y ¼ ∂ ¼ YUYY;pb ¼ ∂ ¼ bUbb: ð9:6Þ Y b b Y We can define the sound speed of perfect fluids as the quantity that relates the variations of the energy density and Unsurprisingly, (9.5) reduces to one of the expressions the pressure along the fluid flow: (9.2) if b or Y are absent from the action. The quantities (9.4), (9.5), and (9.6) allow to express generic variations 0 2 0 of the pressure and energy density as follows, p ¼ cs ρ : ð9:1Þ δ δ 2ρ δ δρ ρ δ −2 δ As we will see in the next section, where we discuss p ¼ pb b þ cb Y Y; ¼ Y Y þ cY pb b; ð9:7Þ dynamical stability, this definition gives the speed of δ ≠ 2δρ propagation of longitudinal phonons, appearing natu- so clearly, p cs . The missing ingredient that allows rally by expanding the action at quadratic order in to turn this into an equality is the variation of the entropy σ fluctuations. For barotropic perfect fluids, (9.1) can be density per particle, ¼ s=n, which we can compute 2 using the dictionary of Table II, obtaining simply computed as cs ¼ dp=dρ using the expressions (4.11),giving ρ δY δb 8 δσ ¼ α Y − c2 ; b Y b b > b Ubb ;U¼ UðbÞ <> Ub 2 2 −ðb=UYÞ ¼ −σ ; for fμ;sg 2 UY α ; : cs ¼ ;U¼ UðYÞ : ð9:2Þ ¼ ð9 8Þ > YUYY 1 > ; for fn; Tg : UX U 2XU ;U¼ UðXÞ X þ XX so that
In the case of Uðb; YÞ, a variation of the pressure is bY δp c2δρ 2 − 2 δσ not uniquely determined by the variation of the energy ¼ s þ α ðcb csÞ : ð9:9Þ density. This is simply because Uðb; YÞ is a nonbaro- tropic fluid; see (4.12). In this case, it is convenient to We recall that in a perfect fluid—for which the entropy define the restricted variations of the pressure with and particle currents are parallel—if one of them is respect to the energy density conserved the other is also conserved, which implies that σ0 ¼ 0; see Sec. III. This is precisely what happens in the μ μ μ 7In the BE case (ϵ ¼ 1), the chemical potential μ is negative case at hand, where Y and J are aligned with u ; see and 0 ≤ α ≤ 1. Table II for their interpretation according to the choice of
025034-10 THERMODYNAMICS OF PERFECT FLUIDS FROM SCALAR … PHYSICAL REVIEW D 94, 025034 (2016) thermodynamic variables. Notice also that σ0 ¼ 0 does effective theory of fluids that we are discussing can not imply in general that σ is vanishing. It is important to be seen as the theory of the propagation of sound emphasize that these results rely on the thermodynamic waves in continuous media [3]. Notice that the fields interpretation of the EFT action, which allows us to πi ¼ Φi − xi are invariant under the combination of identify the currents of entropy and density. Clearly, the a constant translation xi → xi − ci and an internal thermodynamic interpretation is also needed to compute shift Φi → Φi þ ci. Therefore, the πi propagate on a σ and its variation (9.8). It is also worth stressing that homogeneous background and represent the Gold- these arguments are valid at all orders in fluctuations, as stone bosons of broken translations. It is convenient it is clear from the way we have constructed the to split the phonons into a longitudinal component variations in (9.8). and two transverse ones, defined by
πi πi πi πi ∂ π ∂ πi 0 X. PROPAGATION OF PHONONS AND ¼ L þ T; L ¼ i L; i T ¼ : DYNAMICAL STABILITY ð10:7Þ In this section, we study the propagation of the internal degrees of freedom of effective perfect fluids and their The action expanded at quadratic order reads relation to energy and pressure perturbations, as well as the conditions that ensure dynamical stability. The results of 2 1 Sð Þ½b ¼ ðρ þ pÞ this analysis will be related to thermodynamic stability, 2 Z which was discussed in Sec. VIII. For simplicity, we will 4 π_ i π_ i − π_ Δπ_ − 2 Δπ 2 consider perfect fluids living in flat spacetime. Then, the × d x½ T T L L csð LÞ ; dynamics of linear fluctuations are given by the Euler and the continuity equations (EE and CE), which read, ð10:8Þ 8 respectively, 2 where Δ ¼ ∂i∂i and cs can be read in (9.2). The π̈i 0 ρ ∂ i ∂ δ 0 equations of motion (EOMs) are T ¼ and ð þ pÞ tv þ i p ¼ ; ð10:1Þ 2 π̈L − cs ΔπL ¼ 0. Therefore, the transverse modes i do not propagate—their amplitude simply changes ∂tδρ þðρ þ pÞ∂iv ¼ 0; ð10:2Þ linearly in time—due to the conservation of vor- and come from the conservation of Tμν; see (3.2) and (3.4). ticity, which can be traced back to the symmetry i VsDiff. The longitudinal mode propagates with To solve them, an extra relation between δp, δρ and ∂iv needs to be known, which in our case is provided by the speed of sound given by (9.2). The linear pressure δ 2δρ EFT action. The expansion at the second order in the and density perturbations are p ¼ cs ¼ 2 ρ Δπ 2 ρ phonon fields of the three operators of the EFT is given by csð þ pÞ L, and indeed cs ¼ dp=d , as can be seen directly from (4.11). The linear velocity per- 1 1 1 turbation is ui −π_ i, and the divergence of the EE 1 ∂ πi − π_ iπ_ i ∂ πi 2 − ∂ πj ∂ πi ; ¼ b ¼ þ i 2 þ 2 ð i Þ 2 ð i Þð j Þ (10.1) is nothing but the equation for the propagation π ð10:3Þ of L. The CE (10.2) is simply an identity. Stability requires ρ þ p ≥ 0, in order to avoid possibly 2 1 dangerous ghosts, and cs ≥ 0, to avoid exponential Y ¼ 1 þ π_ 0 þ π_ iπ_ i − π_ i∂ π0; ð10:4Þ π ≤ 0 2 i growth of L. The first condition is Ub . If this is 2 satisfied, the constraint cs ≥ 0 is equivalent to the 0 0 2 0 2 ≥ 0 X ¼ −1 − 2_π − ðπ_ Þ − ð∂iπ_ Þ : ð10:5Þ condition for thermodynamic stability Ubb of (8.3). (ii) Let us now consider the irrotational perfect fluid (i) Let us start with UðbÞ and write the scalar fields as UðXÞ. The only scalar that is present in this case is Φ0, which we write in a similar fashion as (10.6): Φi j i πi 0 0 _ ðt; x Þ¼x þ ðt; x~Þ: ð10:6Þ Φ ¼ φðtÞþπ . The EOM for φ is ∂tðUxφÞ¼0 and implies that φ ∝ t (except if ρ → 0, which is a In this expression, πi represent the phonons that are limit we discard). So, we write fluctuations around the static background Φi ¼ xi j Φ0 π0 under the assumption that j∂iπ j ≪ 1. Indeed, the ¼ t þ ; ð10:9Þ
0 8 and expand the action assuming that j∂π j ≪ 1. The In this section, both ∂t and the overdots indicate time derivatives. The background enthalpy per unit of volume, dynamics of the Goldstone boson of the broken time denoted here by ρ þ p, is a constant in Minkowski spacetime. translation is then governed by the action
025034-11 BALLESTEROS, COMELLI, and PILO PHYSICAL REVIEW D 94, 025034 (2016) Z 1 ρ 0 ð2Þ 4 −2 0 2 0 2 þ p ¼ UY > holds, thermodynamic stability S ½X ¼ ðρ þ pÞ d x½cs ðπ_ Þ − ð∂iπ Þ ; 2 2 implies that cs > 0; see (8.3). ð10:10Þ To summarize the results so far, we have seen that for perfect fluids UðbÞ, UðXÞ, and UðYÞ longi- 2 tudinal modes—a single mode for UðbÞ and UðXÞ where cs is given in (9.2). The linear pressure and and two for U Y —propagate with speeds of sound δ −2 π_ 0 ð Þ energy density perturbations are p ¼ pX ¼ c2 dp=dρ, given in (9.2). This reflects that these 2δρ 2 ρ ρ s ¼ cs and, again, cs ¼ dp=d ¼ pX= X from (4.11). fluids are barotropic. We have seen that in these Vi −∂ π0 The velocity perturbation is ¼ i . The EOM three cases, thermodynamic stability plus the null- π0 π̈0 − 2Δπ0 0 for is cs ¼ , which is the CE (10.2), energy condition imply positivity of the speed of whereas the EE (10.1) is now an identity; just the sound, ensuring dynamical stability. opposite of what occurs for UðbÞ. Dynamical (iv) Finally, let us focus on Uðb; YÞ. Expanding as before ρ 2 ≥ 0 ρ ≥ 0 0 stability requires ð þ pÞ=cs and þ p . around Φi ¼ xi and Φ ¼ t, we obtain Thermodynamic stability, see (8.7), demands pre- Z cisely the first of these two conditions. 1 ð2Þ 4 ρ π_ i π_ i − π_ Δπ_ (iii) We will now consider UðYÞ, which contains both S ½b; Y ¼2 d x½ð þ pÞð T T L LÞ types of scalar fields, Φi and Φ0. The appropriate ρ π_ 0 2 − p Δπ 2 − 2c2ρ π_ 0Δπ : background in this case is given by (10.6) and þ Yð Þ bð LÞ b Y L (10.9), as it can be checked using the EOMs. In ð10:13Þ particular, the conservation of (4.10) implies that UY i 0 As before, V Diff gives π̈ 0, and the remaining is a constant, which then requires Φ ∝ t. The s T ¼ EOMs are quadratic action for the phonons around the back- ground is π̈0 − c2Δπ_ ¼ 0; Z b L 1 ρ p π̈ − ρ c2π_ 0 − p Δπ 0; : ð2Þ ρ 4 π_ i π_ i − π_ Δπ_ ð þ Þ L Y b b L ¼ ð10 14Þ S ½Y ¼2 ð þ pÞ d x½ T T L L −2 0 2 0 which give − cs ðπ_ Þ − 2_π ΔπL ; ð10:11Þ 0 2 π_ ¼ c Δπ þ σ0ðx~Þ; where, once more, c2 is given in (9.2). The EOMs b L s ρ 2 derived from the above action, besides π̈i 0, are 2 Ycb T ¼ π̈ − c Δπ ¼ σ0ðx~Þ; ð10:15Þ L s L ρ þ p 0 0 2 π̈L − π_ ¼ 0; π̈ − cs Δπ_ L ¼ 0: ð10:12Þ where σ0ðx~Þ is a generic time-independent function, fixed by initial conditions. The situation is analo- As in the case UðbÞ, π̈i ¼ 0 is a consequence of the T gous to the case UðYÞ with two modes, one that symmetry V Diff. 2 s propagates with velocity c given by (9.5), and a Equating to zero the determinant of the quadratic s second one with ω ¼ 0. The reason is the lack of a form that defines the Lagrangian of (10.11) and quadratic term in (10.13) containing only spatial going to Fourier space, one finds two modes: one π0 ω2 − 2 2 0 derivatives of . with a dispersion relation csk ¼ and a H ω 0 The Hamiltonian density derived from (10.13) is second one with ¼ . This second mode is similar 0 _ πi ω 0 given by H ¼ P0π_ þ PLπ_ L þ P~ · π~T − L, where L is the to the transverse ones, T, which also have ¼ . Notice, as well, that once the equation of motion for Lagrangian density and P0, PL, P~ are the momenta 0 π0 π π πL is solved, the dynamics of π is given by a single conjugate to , L, and ~T, respectively. As a result, time integration constant. As in the case U X , the energy density and 1 _ _ ð Þ H ¼ ½ðp þ ρÞðπ~ · π~ þ ∇~ π_ · ∇~ π_ Þ pressure perturbations depend on π_ 0, i.e. δp ¼ 2 T T L L 2δρ 2 ρ π_ 0 0 2 2 cs ¼ csð þ pÞ , whereas the velocity perturba- þ UYYðπ_ Þ − UbbðΔπLÞ : ð10:16Þ tion is the same as for UðbÞ, i.e. ui ¼ −π_ i. The EE is the first of the EOMs derived from (10.11) and the Thus, H is positive definite when ρ þ p ¼ UY − Ub > 0, i πi CE (10.2) is the other one, since u depends on Ubb < 0 and UYY > 0. Notice that the previous three δρ π0 2 while depends on . In this case dynamical conditions also ensure that cs > 0; see (9.5). The entropy 2 stability requires ρ þ p>0 and cs > 0 as it can perturbation is nonvanishing. Indeed, at linear order, by be checked by imposing that the Hamiltonian using the dictionary entry for Uðb; YÞ relative to ðn; TÞ,we density derived from (10.11) is positive. When have that
025034-12 THERMODYNAMICS OF PERFECT FLUIDS FROM SCALAR … PHYSICAL REVIEW D 94, 025034 (2016) δσ ≡ ρ π_ 0 − 2Δπ ρ σ Yð cb LÞ¼ Y 0ðx~Þ: ð10:17Þ single thermodynamic variable is sufficient for the descrip- tion. Clearly, this is because two independent operators (b It is straightforward to see from (10.15) that although and Y) are needed for a nondegenerate matching with two δσ ≠ 0, it is constant in time, independent thermodynamic variables; and also to describe two independent thermodynamic conserved currents. We ∂tðδσÞ¼0; ð10:18Þ have illustrated the use of the thermodynamic dictionary with a few well-known cases of perfect fluids, such as e.g. as expected for an adiabatic fluid. The conclusion is the Bose-Einstein and Fermi-Dirac gases in the deeply relativ- same when the dictionary entry for Uðb; YÞ relative to ðμ;sÞ istic limit and a van der Waals gas. is used. The analysis of Sec. VIII showed that thermody- We have argued that internal shift symmetries are a namic stability requires Ubb ≤ 0 and UYY ≥ 0. As in the necessary condition for a thermodynamic description by previous cases, thermodynamic stability plus the null- studying the Lagrangian for a single scalar with an energy condition imply dynamical stability. Notice that explicitly broken shift symmetry: UðX; ϕÞ, whose EMT in the cases where an unspecified function appears in the has the form of a perfect fluid, showing that in such a case dictionary, thermodynamic stability implies additional the basic thermodynamic relations are incompatible with constraints on f that cannot be obtained requiring dynami- the equation of motion of the field. cal stability of the phonons. However, in the examples We have also studied the propagation of linearized sound where f is known, such as those of Sec. VII, those waves in flat spacetime and how the Euler and the additional constraints are automatically satisfied. continuity equations describe the dynamics of Goldstone bosons for each kind of perfect fluid. This analysis leads to XI. SUMMARY AND CONCLUSIONS the conclusion that thermodynamic stability plus the null- energy condition, ρ þ p ≥ 0, ensure dynamical stability. We have studied thermodynamic interpretation of the This holds true for the four possible types of effective EFT of perfect fluids, obtaining a dictionary summarized in perfect fluids. The same analysis shows that the fluid Table II, completing and extending the results of [4,5].We described by Uðb; YÞ, being in general nonbarotropic, can have established the correspondence between the funda- support nonvanishing entropy per particle perturbations but mental thermodynamic variables needed to describe simple is nonetheless adiabatic. systems and the EFT operators that configure the four types of perfect fluids that are allowed in the theory. Each entry of the dictionary (described by Table II) corresponds to a ACKNOWLEDGMENTS specific operator content in the EFT. The interpretation of We thank B. Bellazzini, L. Hui, I. Sawicki, and S. the EFT master function U—the Lagrangian—as a thermo- Sibiryakov for useful discussions. L. P. thanks S. Ciuchi for dynamic potential is determined by the EMT. For the interesting discussions and suggestions. The work of G. B. effective perfect fluids depending on a single scalar is funded by the European Union’s Horizon 2020 research operator, the master function U represents either the energy and innovation program under the Marie Skłodowska- density (for UðbÞ) or the pressure (for UðYÞ and UðXÞ). For Curie Grant Agreement No. 656794. G. B. thanks the these cases, the thermodynamic potentials F and I appears CERN Theoretical Physics Department for hospitality as specific limits of the free function f of Table II. For the while this work was done. L. P. thanks the Institute de Lagrangian Uðb; YÞ, the master function corresponds to a Physique Théorique IPhT CEA-Saclay for hospitality. We Legendre transformation of the energy density (or the also thank the Galileo Galilei Institute for Theoretical pressure) to another thermodynamic potential, such as Physics for hospitality and the Instituto Nazionale di the free energy F ¼ ρ − Ts or to the potential I ¼ ρ − μn. Fisica Nucleare (INFN) for support during the completion A full thermodynamic correspondence, allowing to iden- of this work. tify simultaneously an independent and conserved particle (or charge) number current and a conserved entropy density APPENDIX: A SYSTEMATIC APPROACH TO current is only possible for perfect fluids described by the THE THERMODYNAMIC DICTIONARY Lagrangian Uðb; YÞ. This action is invariant under the internal symmetry group VsDiff × Ts of spatial volume Any simple thermodynamic system is described by at preserving diffeomorphisms and time redefinitions that most six variables: ζ ¼fs; T; n; μ; ρ;pg, of which only two depend on the spatial fields. The fact that Uðb; YÞ is the are independent. For the EFT of perfect fluids, ρ and p are only perfect fluid Lagrangian that is chosen by a (continu- given as functions of at most two operators among ous) symmetry (assuming four scalars), highlights it even O ¼fb; Y; Xg. Let us choose two independent variables more as the most complete effective description of perfect (other than ρ and p) from the list ζ and denote them z1 and fluids; see also [5]. The other perfect fluids, indicated as z2. These can be, for instance, fn; sg or fμ;Tg; see Table I. UðbÞ, UðYÞ and UðXÞ in Table II, can also be given Since all the other four variables contained in ζ are, by thermodynamic interpretations, but only in limits where a assumption, functions of z1 and z2, we can formally write
025034-13 BALLESTEROS, COMELLI, and PILO PHYSICAL REVIEW D 94, 025034 (2016) ζ Z Z derivatives of U. One can easily keep track of the number ðz1;z2Þ¼f|fflfflfflffl{zfflfflfflffl}z1;z2g × f|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}1ðz1;z2Þ; 2ðz1;z2Þg of derivatives entering in each quantity using a counting Indep var Dependent variables parameter ϵ, such that ρ O O × |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}f ½ ðz1;z2Þ ;p½ ðz1;z2Þ g; ðA1Þ Variables from the EMT UðnÞðOÞ → ϵnUðnÞðOÞ;
n1;n2 n1;n2 where we have explicitly taken into account that the Zð Þ → ϵ1þn1þn2 Zð Þ 1;2 ðz1;z2Þ 1;2 ðz1;z2Þ; operators O must be functions of z1 and z2, and we have Oðn1;n2Þ → ϵ0Oðn1;n2Þ denoted by Z1 and Z2 the two (dependent) variables that ðz1;z2Þ ðz1;z2Þ: ðA2Þ are not ρ and p. The thermodynamic relations (2.3), (2.5), Z and (2.7) constrain the dependence of 1;2ðz1;z2Þ and As an example, let us take the case Uðb; YÞ with fz1;z2g¼ Oðz1;z2Þ on the chosen independent variables z1 and z2. fn; Tg as independent variables. From (4.12), we have that The thermodynamic dictionary for the EFT of perfect fluids ρ ¼ YUY − U and p ¼ U − bUb. Then, fZ1; Z2g¼ can be derived, requiring that fsðn; TÞ; μðn; TÞg and O ¼ðY;bÞ are also functions of (i) It has to be valid for any master function U. n and T. The Euler relation (2.5) reads (ii) The operators O are independent of U and its derivatives. ϵ − ϵ μ Then, any thermodynamic constraint involving the deriv- ðYUY bUbÞ¼ ½Tsðn; TÞþn ðn; TÞ ; ðA3Þ atives UðnÞ must hold irrespectively of the form of UðnÞ. Moreover, from the Euler relation (2.5), we see that the consistently with (A2). Solving for sðn; TÞ and inserting it dependent variables Z1;2ðz1;z2Þ will depend on first in (2.3), or equivalently in (2.7), we get
2 ϵ − 2 − − μ ϵ Tf ½bð bnUb YnUbYÞ n n þ YnUYgdn 2 ϵ − 2 − − μ ϵ μ − 0 þf T½bð bTUb YTUbYÞ n T þ ðbUb þ n YUY þ TYTUYÞgdT ¼ : ðA4Þ
The coefficients of dn and dT must vanish independently, for all U and order by order in ϵ. At order ϵ, we get ϵYn ¼ 0 and 2 2 ϵðbUb þ nμ − YUY þ TYTUYÞ¼0. Differentiating these equations with respect to n and T, we obtain ϵ Ynn ¼ ϵ YnT ¼ 0 and
2 2 2 ϵ μ ϵ − − 2 ϵ − − nn ¼ nbn½ðY TYTÞUbY bUb þ ½ðb nbnÞUb þðTYT YÞUY ; 2 2 ϵ μ ϵ − 2 − − 2 − ϵ 2 n T ¼ ½ bbTUb þðYbT ðTbT þ bÞYTÞUbY þ YTðY TYTÞUY ðbTUb þ TYT UYÞ:
New terms of order ϵ appear and they have to vanish, leading to ðb − nbnÞUb þðTYT − YÞUY ¼ 0 and 2 0 bTUb þ TYT UY ¼ . The solution of these last equations (independent from the form of U)isb ¼ nbn, Y ¼ TYT, bT ¼ YTT ¼ 0 and, therefore,
b ¼ n; Y ¼ T; μ ¼ −Ub;s¼ UY: ðA5Þ
One can now check that all the thermodynamic relations are satisfied at all orders in ϵ. The solution (A5) gives the entry of Table II relative to Uðb; YÞ with ðn; TÞ as independent variables in our dictionary. All the other entries can be derived in the same way.
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